Knot Physics is a unification theory that describes all physical phenomena using a branched spacetime manifold embedded in a Minkowski 6-space. Manifold topology and geometry result in particles: elementary fermions are topological defects in the manifold, and elementary bosons are geometric features of the manifold. In addition, the spacetime manifold has constraints but is under-constrained; thus, the behavior of the spacetime manifold is random with respect to its degrees of freedom. Entropy maximization results in the dynamics of all physical phenomena, including quantum mechanics and all forces.

## Fundamentals

### Components and Constraints

The spacetime manifold is embedded in a larger space. The components of that embedding are:

A Minkowski 6-space, \(\Omega\)

A 4-dimensional branched spacetime manifold, \(M\), which is embedded in \(\Omega\)

The metrics are:

A metric on \(\Omega\), which is \(\eta_{\mu\nu} = diag(1,-1,-1,-1,-1,-1) \)

A metric \(\bar \eta_{\mu\nu}\) on \(M\), which is the metric induced on \(M\) by its embedding in \(\Omega\)

A metric \(g_{\mu\nu}\) on \(M\) that constrains the geometry and branching of \(M\)

The constraints on \(M\) are constraints on the metric \(g_{\mu\nu}\). For example, \(M\) is assumed to be constrained such that the metric \(g_{\mu\nu}\) is Ricci flat.

### Topology and Fermions

The spacetime manifold, \(M\), is embedded in a Minkowski space. This embedding tightly constrains the type of topology change that is possible on \(M\). For example, all cobordism on \(M\) must be trivial, implying that no change of topology is possible by cobordism. The constraints on \(M\) are consistent, however, with the possibility of passing through a singular state that results in pairs of topological defects on \(M\) that have homeomorphism class \( \mathbb{R}^3 \# (S^1 \times P^2)\). In Knot Physics, a topological defect \( \mathbb{R}^3 \# (S^1 \times P^2)\) is referred to as a “knot.”

This approach enables a geometric description of fermions.

The creation of pairs of knots corresponds to the creation of

**particle–antiparticle pairs**.The knots are the

**elementary fermions**.The knots allow multiple embeddings, and the different knot embeddings correspond to the different

**generations of elementary fermions**.Two or more knots can link to each other, and linked knots are

**quarks**.

### Entropy Maximization and Dynamics

Because the spacetime manifold is under-constrained, it has degrees of freedom. Subject to its constraints, \(M\) maximizes entropy. All of the dynamics of \(M\) results from that entropy maximization.

**Quantum mechanics**results from random recombination of the branches of \(M\). Quantum information is encoded in the branches of \(M\). The branch structure is non-local and explains**non-local quantum entanglement**.Maximization of the entropy of the geometry of \(M\) results in all the forces:

**electromagnetism**,**weak force**,**strong force**, and**gravity**. The force carrier bosons are geometric features of the manifold that correspond to interactions between elementary fermions.

### Calculation

The geometry of an electron knot allows a calculation of its spin angular momentum as a function of charge. Because the angular momentum is \(ℏ/2\), the calculation allows a comparison of electron charge to Planck’s constant, which gives a **derivation of the fine structure constant**. The estimate is accurate to 0.1%. Calculation of additional Feynman diagrams may result in additional accuracy.

## Beyond the Standard Model

### Dark Energy

The spacetime manifold is embedded in a larger space, and expansion of the manifold corresponds to expansion as an embedding in the larger space. The velocity of that expansion contributes to the red shift from distant astrophysical sources. That additional contribution may explain some, or all, of the data that is currently used to infer the existence of dark energy.

### Dark Matter

The branches of the spacetime manifold are constrained by a conserved quantity, called branch weight, \(w = (-\det(g))^{1/2}\). The distribution of branch weight affects spacetime curvature. Branch weight distribution can therefore explain curvature that is attributed to dark matter without the need for additional particles.